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Theorem mdegval 24043

Description: Value of the multivariate degree function at some particular polynomial. (Contributed by Stefan O'Rear, 19-Mar-2015.) (Revised by AV, 25-Jun-2019.)

**Hypotheses**
Ref	Expression
mdegval.d	⊢ 𝐷 = (𝐼 mDeg 𝑅)
mdegval.p	⊢ 𝑃 = (𝐼 mPoly 𝑅)
mdegval.b	⊢ 𝐵 = (Base‘𝑃)
mdegval.z	⊢ 0 = (0_g‘𝑅)
mdegval.a	⊢ 𝐴 = {𝑚 ∈ (ℕ₀ ↑_𝑚 𝐼) ∣ (^◡𝑚 “ ℕ) ∈ Fin}
mdegval.h	⊢ 𝐻 = (ℎ ∈ 𝐴 ↦ (ℂ_fld Σ_g ℎ))

**Assertion**
Ref	Expression
mdegval	⊢ (𝐹 ∈ 𝐵 → (𝐷‘𝐹) = sup((𝐻 “ (𝐹 supp 0 )), ℝ^*, < ))

Distinct variable groups: 𝐴,ℎ 𝑚,𝐼 0 ,ℎ Allowed substitution hints: 𝐴(𝑚) 𝐵(ℎ,𝑚) 𝐷(ℎ,𝑚) 𝑃(ℎ,𝑚) 𝑅(ℎ,𝑚) 𝐹(ℎ,𝑚) 𝐻(ℎ,𝑚) 𝐼(ℎ) 0 (𝑚)

Proof of Theorem mdegval Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq1 6803	. . . 4 ⊢ (𝑓 = 𝐹 → (𝑓 supp 0 ) = (𝐹 supp 0 ))
2	1	imaeq2d 5606	. . 3 ⊢ (𝑓 = 𝐹 → (𝐻 “ (𝑓 supp 0 )) = (𝐻 “ (𝐹 supp 0 )))
3	2	supeq1d 8512	. 2 ⊢ (𝑓 = 𝐹 → sup((𝐻 “ (𝑓 supp 0 )), ℝ^, < ) = sup((𝐻 “ (𝐹 supp 0 )), ℝ^, < ))
4		mdegval.d	. . 3 ⊢ 𝐷 = (𝐼 mDeg 𝑅)
5		mdegval.p	. . 3 ⊢ 𝑃 = (𝐼 mPoly 𝑅)
6		mdegval.b	. . 3 ⊢ 𝐵 = (Base‘𝑃)
7		mdegval.z	. . 3 ⊢ 0 = (0_g‘𝑅)
8		mdegval.a	. . 3 ⊢ 𝐴 = {𝑚 ∈ (ℕ₀ ↑_𝑚 𝐼) ∣ (^◡𝑚 “ ℕ) ∈ Fin}
9		mdegval.h	. . 3 ⊢ 𝐻 = (ℎ ∈ 𝐴 ↦ (ℂ_fld Σ_g ℎ))
10	4, 5, 6, 7, 8, 9	mdegfval 24042	. 2 ⊢ 𝐷 = (𝑓 ∈ 𝐵 ↦ sup((𝐻 “ (𝑓 supp 0 )), ℝ^*, < ))
11		xrltso 12179	. . 3 ⊢ < Or ℝ^*
12	11	supex 8529	. 2 ⊢ sup((𝐻 “ (𝐹 supp 0 )), ℝ^*, < ) ∈ V
13	3, 10, 12	fvmpt 6426	1 ⊢ (𝐹 ∈ 𝐵 → (𝐷‘𝐹) = sup((𝐻 “ (𝐹 supp 0 )), ℝ^*, < ))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1631 ∈ wcel 2145 {crab 3065 ↦ cmpt 4864 ^◡ccnv 5249 “ cima 5253 ‘cfv 6030 (class class class)co 6796 supp csupp 7450 ↑_𝑚 cmap 8013 Fincfn 8113 supcsup 8506 ℝ^*cxr 10279 < clt 10280 ℕcn 11226 ℕ₀cn0 11499 Basecbs 16064 0_gc0g 16308 Σ_g cgsu 16309 mPoly cmpl 19568 ℂ_fldccnfld 19961 mDeg cmdg 24033

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4905 ax-sep 4916 ax-nul 4924 ax-pow 4975 ax-pr 5035 ax-un 7100 ax-cnex 10198 ax-resscn 10199 ax-1cn 10200 ax-icn 10201 ax-addcl 10202 ax-addrcl 10203 ax-mulcl 10204 ax-mulrcl 10205 ax-mulcom 10206 ax-addass 10207 ax-mulass 10208 ax-distr 10209 ax-i2m1 10210 ax-1ne0 10211 ax-1rid 10212 ax-rnegex 10213 ax-rrecex 10214 ax-cnre 10215 ax-pre-lttri 10216 ax-pre-lttrn 10217 ax-pre-ltadd 10218 ax-pre-mulgt0 10219

This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4227 df-pw 4300 df-sn 4318 df-pr 4320 df-tp 4322 df-op 4324 df-uni 4576 df-int 4613 df-iun 4657 df-br 4788 df-opab 4848 df-mpt 4865 df-tr 4888 df-id 5158 df-eprel 5163 df-po 5171 df-so 5172 df-fr 5209 df-we 5211 df-xp 5256 df-rel 5257 df-cnv 5258 df-co 5259 df-dm 5260 df-rn 5261 df-res 5262 df-ima 5263 df-pred 5822 df-ord 5868 df-on 5869 df-lim 5870 df-suc 5871 df-iota 5993 df-fun 6032 df-fn 6033 df-f 6034 df-f1 6035 df-fo 6036 df-f1o 6037 df-fv 6038 df-riota 6757 df-ov 6799 df-oprab 6800 df-mpt2 6801 df-of 7048 df-om 7217 df-1st 7319 df-2nd 7320 df-supp 7451 df-wrecs 7563 df-recs 7625 df-rdg 7663 df-1o 7717 df-oadd 7721 df-er 7900 df-map 8015 df-en 8114 df-dom 8115 df-sdom 8116 df-fin 8117 df-fsupp 8436 df-sup 8508 df-pnf 10282 df-mnf 10283 df-xr 10284 df-ltxr 10285 df-le 10286 df-sub 10474 df-neg 10475 df-nn 11227 df-2 11285 df-3 11286 df-4 11287 df-5 11288 df-6 11289 df-7 11290 df-8 11291 df-9 11292 df-n0 11500 df-z 11585 df-uz 11894 df-fz 12534 df-struct 16066 df-ndx 16067 df-slot 16068 df-base 16070 df-sets 16071 df-ress 16072 df-plusg 16162 df-mulr 16163 df-sca 16165 df-vsca 16166 df-tset 16168 df-psr 19571 df-mpl 19573 df-mdeg 24035

This theorem is referenced by: mdegleb 24044 mdeglt 24045 mdegldg 24046 mdegxrcl 24047 mdegcl 24049 mdeg0 24050 mdegvsca 24056 deg1val 24076

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